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Theorem efrirr 5658
Description: A well-founded class does not belong to itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
efrirr ( E Fr 𝐴 → ¬ 𝐴𝐴)

Proof of Theorem efrirr
StepHypRef Expression
1 frirr 5654 . . 3 (( E Fr 𝐴𝐴𝐴) → ¬ 𝐴 E 𝐴)
2 epelg 5582 . . . 4 (𝐴𝐴 → (𝐴 E 𝐴𝐴𝐴))
32adantl 483 . . 3 (( E Fr 𝐴𝐴𝐴) → (𝐴 E 𝐴𝐴𝐴))
41, 3mtbid 324 . 2 (( E Fr 𝐴𝐴𝐴) → ¬ 𝐴𝐴)
54pm2.01da 798 1 ( E Fr 𝐴 → ¬ 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wcel 2107   class class class wbr 5149   E cep 5580   Fr wfr 5629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-fr 5632
This theorem is referenced by:  tz7.2  5661  ordirr  6383
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